The Classification Conjecture for contraction algebras of smooth threefold flops

Let $\EuScript{X}_1\to\operatorname{Spec}\EuScript{R}_1$ and $\EuScript{X}_2\to\operatorname{Spec}\EuScript{R}_2$ be $3$-fold flopping contractions, where $\EuScript{X}_i$ are smooth and $\EuScript{R}_i$ are complete local. Denote their corresponding contraction algebras by $\mathrm{A}_{\operatorname{con}}$ and $\mathrm{B}_{\operatorname{con}}$, respectively.

The Classification Conjecture. The complete local cDV singularities are isomorphic if and only if their contraction algebras have equivalent bounded derived categories:

$$\EuScript{R}_1\cong\EuScript{R}_2\iff D^b(\operatorname{mod}\mathrm{A}_{\operatorname{con}})\simeq D^b(\operatorname{mod}\mathrm{B}_{\operatorname{con}}).$$

This asserts that the algebraic structure of the contraction algebra classifies the underlying cDV singularity in the smooth threefold flopping setting. The source records evidence and enhanced versions known in several cases, but does not give a resolution of the conjecture in full generality.